Question:
Express $\left(\frac{1}{3}+3i\right)^{3}$ in the form of $a +ib$.
Express $\left(\frac{1}{3}+3i\right)^{3}$ in the form of $a +ib$.
Updated On: Jul 6, 2022
- $1+26i$
- $\frac{-242}{27}+26i$
- $\frac{-242}{27}-26i$
- $-1+26i$
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The Correct Option is C
Solution and Explanation
$\left(\frac{1}{3}+3i\right)^{3} =\left(\frac{1}{3}\right)^{3}+3\left(\frac{1}{3}\right)^{2} \left(3i\right)+3\left(\frac{1}{3}\right)\left(3i\right)^{2}+\left(3i\right)^{3}$
$=\frac{1}{27}+i+9\left(-1\right)+27\left(-i\right)$
$=\frac{1}{27}-9-26i$
$=\frac{1-243}{27}-26i$
$=\frac{-242}{27}-26i$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
